Part 1: Always
Part 2: Either Always or Never (see Explanation for details)
The three bisectors (also known as medians) of a right triangle will always intersect at the center of mass of the triangle. The center of mass is like the point at which you could balance the shape on the tip of your finger without it falling off.
Since right triangles (and ALL triangles, for that matter) are convex shapes, the center of mass MUST be inside the shape. It wouldn't make any sense for the balancing point of the triangle to be outside of the triangle, would it?
Therefore, the bisectors of a right triangle will ALWAYS intersect inside the triangle.
The three perpendicular bisectors of a triangle always intersect at a point known as the circumcenter. This point is the center of the circle which the right triangle can be inscribed in.
One of the principles of geometry is that the hypotenuse of a right triangle inscribed in a circle will always be a DIAMETER of the circle. This means that the center of the circle is located right in the middle of the hypotenuse, no matter what the other two sides of the right triangle are.
And since the center of the circle is where the three perpendicular bisectors intersect, we can also conclude that the three perpendicular bisectors intersect in the middle of the hypotenuse.
However, this is where the wording of the problem is a bit tricky. The intersection is directly ON one of the sides. Is that still considered inside the triangle? If the triangle is considered the set of points bounded by its edges, then yes, technically the intersection is still within that set of points, but the word "inside" could be asking for "Is the intersection between the three sides of the triangle?", in which case the answer is no (since it is directly ON the sides).
If your teacher (or the problem) has any guidelines for what is considered "inside" the triangle, you should use those to answer the question; I have given you everything I can, but in the end, the problem is worded poorly, so it's not clear what's being asked for.